Complexity emerges when many simple units interact: studying the individual components does not explain the emergence of collective properties
Simple examples:
phases of matter
magnetism, superconductivity
emergence of order from disorder
Also beyond physics:
formation of biological molecules
ecological networks
economic structures (markets, currencies etc.)
Complex mixtures, spintronic magnetisation, ecological networks and Hong Kong stock market
Beyond the usual phases of matter
We sometimes oversimplify…
Extreme examples of phases of matter
Beyond the usual phases of matter
We sometimes oversimplify…
Complex fluids and soft condensed matter
Soft condensed matter includes assembllies of colloids, polymers, surfactants, and biological macromolecules and much more.
Term due to Pierre-Gilles de Gennes (Nobel 1991)
Often, many ofthese are also referred to as complex fluids.
These materials are easily deformed and show complex, disordered structures.
Their properties are determined by a balance between energy and entropy.
Thermal fluctuations play a major role in their behavior.
Understanding them requires statistical mechanics .
Thermal fluctuations and entropy
Helmholtz Free energy per particle f = u-Ts
Entropy approximately counts the number of arrangements per particle s = \dfrac{k_B}{N} \ln{\Omega}. For hundreds of arrangements per particle one has s=O(1) k_B
Fluctuations of the internal energy are on the same scale as thermal fluctuations:
\boxed{\Delta u \sim k_B T}
where k_B is the Boltzmann constant and \Delta u indicates standard deviations from the average internal energy.
Thermal fluctuations: single particle in a doyble well
viewof params = Inputs.form({T: Inputs.range([0.,4], {label:"Temp.",value:1.0,step:0.1,}),g: Inputs.range([-2,2], {label:"Tilt",value:0,step:0.1}),reset: Inputs.button("Reset simulation")})// Simulation parametersdt =0.005kB =1gamma =1// Double well potential: V(x) = x^4 - 2x^2 + g*xV = (x, g) => x**4-2*x**2+ g*xforce = (x, g) =>-(4*x**3-4*x + g)// Live simulation with temperature controlsimulationState = {let x =0let positions = [0]let step =0while (true) {if (params.reset) { x =0 positions = [0] step =0 }let f =force(x, params.g)let noise =Math.sqrt(2* kB * params.T* dt / gamma) * (Math.random() -0.5) *2 x += (f / gamma) * dt + noise positions.push(x)// Keep only last 500 points for performanceif (positions.length>500) { positions = positions.slice(-500) }yield {x,positions: [...positions],step: step++}awaitnewPromise(resolve =>setTimeout(resolve,10)) }}// PlotPlot.plot({width:700,height:400,grid:true,x: {domain: [-2.5,2.5],label:"Position"},y: {domain: [-2,3],label:"Potential Energy"},marks: [// Potential curve Plot.line(Array.from({length:200}, (_, i) => {let x =-2.5+5* i /199return {x,y:V(x, params.g)} }), {x:"x",y:"y",stroke:"red",strokeWidth:2} ),// Particle trajectory (trail) Plot.line( simulationState.positions.map((x, i) => ({x,y:V(x, params.g),step: i})), {x:"x",y:"y",stroke:"blue",strokeWidth:1,opacity:0.5} ),// Current particle position Plot.dot([{x: simulationState.x,y:V(simulationState.x, params.g) }], {x:"x",y:"y",fill:"blue",r:6,stroke:"white",strokeWidth:2}) ]})
Systems and definitions
Elementary constituents and energy scales
Soft matter systems are made of many parts
The assembly of these many parts can be easily deformed.
Interactions between these parts are weak compared to thermal or mechanical forces.
Hard condensed matter:
Basic units: atoms
Strong interactions (0.1–10 eV)
Covalent/ionic bonds
Focus on low temperatures
Soft matter:
Basic units: molecular aggregates
Weak interactions (0.001–0.2 eV)
Van der Waals, hydrogen bonds
1 k_B T \approx 0.025 eV (room temperature)
Coarse graining
Soft matter interactions are mainly electrostatic.
Atomistic details are often unimportant for macroscopic properties.
Coarse-graining simplifies models by focusing on key features:
deliberate selection of what matters
systematic integration of a number of degrees of freedom
Example: The oxDNA model represents DNA as a chain of coarse grained units, much larger than the atoms.
Atomistic DNA structure
Coarse graining
Soft matter interactions are mainly electrostatic.
Atomistic details are often unimportant for macroscopic properties.
Coarse-graining simplifies models by focusing on key features:
deliberate selection of what matters
systematic integration of a number of degrees of freedom
Example: The oxDNA model represents DNA as a chain of coarse grained units, much larger than the atoms.
oxDNA model: (a) Base structure on one strand; (b) planarity of the bonding; (c) an example of the resulting double strand.
Classes of systems
In our exploration of soft matter we will focus on six main classes of systems which display different physics:
colloidal dispersions
polymeric systems
liquid crystals
surfactant aggregates
arrested systems
active matter
Colloidal assemblies, polymers, surfactants, liuqid crystals, glasses and bacteria.
Colloidal dispersions
Colloidal dispersions: small particles (nano–micrometer) suspended in a solvent.
Spherical colloids are common, but many shapes and interactions exist.
Behave as “big atoms”: show Brownian motion, phase transitions, and can form ordered structures.
Larger size and slower dynamics make them ideal for direct observation of phenomena like:
crystallisation
glass formation
gel formation
(A) Solvent collisions cause random forces on particles. (B) Particles execute random walks on larger timescales. (C) Hard spheres crystallize at high density (right) due to reduced available space compared to fluid phase (left). From Manohran, Science (2015)
Polymeric systems
Polymers are long-chain macromolecules made of repeating monomers.
Their properties result from a balance of entropy and energy.
Two main types: synthetic polymers (e.g., plastics) and biopolymers (e.g., DNA, proteins).
Entanglement: chains cannot cross, leading to unique mechanical behavior.
Polymer entanglement, from Likhtman and Ponmurugan, Macromolecules (2014)
Liquid crystals
Liquid crystals form when anisotropic soft matter units (e.g., rod-like or disk-shaped molecules) pack densely, leading to partial order—intermediate between liquids and crystals.
Continuum free energy theories describe liquid crystals by considering the symmetry of their order parameters.
Liquid crystals are crucial in technologies like liquid crystal displays (LCDs).
The International Union of Pure and Applied Chemistry (IUPAC) defines
colloidal: The term refers to a state of subdivision, implying that the molecules or polymolecular particles dispersed in a medium have at least in one direction a dimension roughly between 1 nm and 1 \mum , or that in a system discontinuities are found at distances of that order.
Note
The definition has little to do with the chemistry of the polymolecules (i.e. aggregates), but essentially is determined by the size.
Why the size?
Colloidal scale
Particle of size R (micrometric) in medium with constant collisions at thermal energy\sim 1 k_B T
Compare the energy with the potential energy of settling over a length R
Obtain a nondimensional number the gravitational Péclet number
\mathrm{Pe}_g=\frac{\Delta m g R}{k_B T}
Example
For a milk protein particle (casein micelle) in water:
Any measurement in physics has an observation timet_{\rm obs}
Thermal systems have some degree of memory and hence an intrinsic timescale \tau (relaxation time)
When \tau\ll t_{\rm obs} we can take time-averages and consider the system in a stable steady state.
In the absence of net currents, this is an equilibrium state
A colloidal dispersion is stable if it is able to remain dispersed and Brownian time much longer than t_{\rm obs}.
Instability leads to aggregation
Destabilisation
Milk is an emulsion where droplets of fat are dispersed in water and stabilised by proteins.
If lemon juice is added, the dispersion medium (water) changes, the pH drops, and the emulsion is destabilised.
The interactions change → separation of curds (solid) and whey (liquid).
Curds and whey resulting from the destabilisation of milk, a colloidal emulsion. (Wikimedia)
Which interactions?
At the colloidal scale: gravity and electrostatics (eventually magnetostatics).
Electrostatics forces are the main microscopic forces: colloids are typically charged
Separation of length and timescales: colloids are much larger and slower than the atomic or molecular constituents of the surrounding fluid.
Colloid–colloid interactions result from the collective effect (sum or average) of many microscopic interactions, evaluated over times much longer than microscopic relaxation times.
When we focus on colloidal scales, we integrate out the microscopic degrees of freedom (coarse-graning)
Coarse-grained interactions are then obtained and are called effective interactions.